MaplePrimes Questions

I'm attempting to find the eigenvectors of a matrix without using the eigenvector function.

The matrix in question is a covariance matrix:

XCov:=Matrix([[4048/5, -817/5, -122/5], [-817/5, 921/10, -1999/10], [-122/5, -1999/10, 8341/10]]);

I've already found the eigenvalues by solving for lambda:

 det := Determinant(XCov-lambda*IdentityMatrix(3));
  lambda := solve(det=0.0, lambda);

(Yes I'm reusing the eigenvalue variable for the set of eigenvalues once they've been found­čśĆ)

Anyway, I've now set up the first eigenvector I want to find as:
e1 := Vector([e11,e12,e13]);


Now, the equation to find this first eigenvector is XCov . e1 = lambda[1] . e1
I first tried putting whats on the left in a variable called eigscale(what the vector is translated to by the matrix):

eigscale := Multiply(XCov,e1);
Which returns a vector:
eigscale = [(4048/5)*e11-(817/5)*e12-(122/5)*e13,
                  -(817/5)*e11+(921/10)*e12-(1999/10)*e13,
                  -(122/5)*e11-(1999/10)*e12+(8341/10)*e13]

Each component of this vector must equate to the corresponding component in the right vector:

lambda[1]*e1 = [7.943520930*e11, 7.943520930*e12, 7.943520930*e13]

At first I tried setting these vectors equal to each other and using a solve but of course it didnt like the equations being in a vector format. So I then seperated out each equation and gave the solve function a system of equations as it expects:

solve(eigscale[1] = lambda[1]*e1[1], eigscale[2] = lambda[1]*e1[2], eigscale[3] = lambda[1]*e1[3], [e11,e12,e13]);

But again, solve fails to solve them. The reason this time(I believe) is because it can't find an exact value for e11, e12 & e13.
When solving for an eigenvector we get
e11 = e11,
e12 = Ae11,
e13 = Be11 + Ce12

I was wondering if there was a way to do a partial solve to find the components in terms of each other?

Failing that, I'm aware I can do it manually through row operations but I believe that would require changing the format so that each equation is a component of a single vector:
eigsolve := Vector([eigscale[1] = lambda[1]*e1[1], eigscale[2] = lambda[1]*e1[2], eigscale[3] = lambda[1]*e1[3]]);

Since row operations cannot be performed on a equation of vectors (again, I believe).

Help appreciated!

Hello Everyone,

 

I have been working on a multi variable expression. I would like to have the intervals where the function is monotonically increasing. I am trying to study any available method for multivariable expressions.

I came across various papers and sites which are explicitly mentioned single variable equations. finding out the critical points and studying the sign of the first derivative. Same cannot be applied for the multi variable expression.

 https://www.math24.net/monotonic-functions/

above link explain for single variable functions. I would be grateful if someone could explain me a method or idea  which helps me out in solving for multivariable functions

 

Thanks a lot in advance

restart;
Digits := 25;

# Setup.
de := diff( x(t), t ) = a * y(t), diff( y(t), t ) = b * x(t);
ic := x(0) = 1, y(0) = 0;
sol := dsolve( { de, ic } ):

# Sample values for specific parameter values.
a0, b0 := 1.0, 0.25:
T := evalf( [ seq( i, i=1..5 ) ] ):
X := [ seq( eval['recurse']( x(t), [ op( sol ), a = a0, b = b0 ] ), t=T ) ]:
Y := [ seq( eval['recurse']( y(t), [ op( sol ), a = a0, b = b0 ] ), t=T ) ]:

# Numerical solution.
sol := dsolve( { de, ic }, 'numeric', 'range'=min(T)..max(T), 'abserr'=1e-15, 'maxfun'=0, 'parameters'=[a,b] ):

# Procedure to return values for specific time and parameter values.
u := proc( t, a, b )
	sol( 'parameters' = [ ':-a' = a, ':-b' = b ] ):
	try
		return eval( [ x( ':-t' ), y( ':-t' ) ], sol( t ) ):  
	catch:
		return 10000:
    end try:
end proc:

# Procedure to return objective function.
r := proc( a, b )
	local A, t, p, q, x, y:
	A := ListTools:-Flatten( [ seq( u( t, a, b ), t=T ) ] -~ zip( (x,y) -> [x,y], X, Y ) ):
	return foldl( (p,q) -> p + q^2, 0, op(A) ):
end proc:

# Find parameter values for bet fit.
Optimization:-Minimize( 'r'(a,b), a=-10..10, b=-10..10 );

I was provided the code above by Maple support, to fit a model to data using the Optimization package. I have tried to adapt this to a situation where one might only have the X variable data and so the model should only fit to that data.  I did this by deleting the , y( ':-t' ) in the try catch statement and then having the zip statement read as zip( (x) -> [x], X).

When I try to run that code, I get the error:

Error, (in Optimization:-NLPSolve) invalid input: zip uses a 3rd argument, b, which is missing

Anybody have an idea how to solve this?

 

Jo

Hi,

I try to define the action of projectors of two discrete basis onto a general state. This works as expected when I define the projector by myself. However, when using the "Projector" command, I get a not fully simplified result; see below. It seems like there is a confusion with dot/tensor product.  Can somoeone help?

Best,

Henrik


 

restart; with(Physics)

Setup(hilbertspaces = {{A, alpha}, {B, beta}}, quantumbasisdimension = {A = 1 .. N[a], B = 1 .. N[b]}, quantumdiscretebasis = {A, B, alpha, beta}, bracketrules = {%Bracket(Bra(A, i), Ket(Psi)) = Ket(beta, i), %Bracket(Bra(A, i), Ket(alpha, j)) = C[i, j], %Bracket(Bra(B, i), Ket(Psi)) = Ket(alpha, i), %Bracket(Bra(B, j), Ket(beta, i)) = C[i, j]})

[bracketrules = {%Bracket(%Bra(A, i), %Ket(Psi)) = Physics:-Ket(beta, i), %Bracket(%Bra(A, i), %Ket(alpha, j)) = C[i, j], %Bracket(%Bra(B, i), %Ket(Psi)) = Physics:-Ket(alpha, i), %Bracket(%Bra(B, j), %Ket(beta, i)) = C[i, j]}, disjointedspaces = {{A, alpha}, {B, beta}}, quantumbasisdimension = {A = 1 .. N[a], B = 1 .. N[b]}, quantumdiscretebasis = {A, B, alpha, beta}]

(1)

``

proj := Sum(Sum(Ket(A, i).Bra(A, i).Ket(B, j).Bra(B, j), i = 1 .. N[a]), j = 1 .. N[b])

Sum(Sum(Physics:-`*`(Physics:-Ket(A, i), Physics:-Ket(B, j), Physics:-Bra(A, i), Physics:-Bra(B, j)), i = 1 .. N[a]), j = 1 .. N[b])

(2)

proj2 := Projector(Ket(A, i)).Projector(Ket(B, i))

Physics:-`*`(Sum(Physics:-`*`(Physics:-Ket(A, i), Physics:-Bra(A, i)), i = 1 .. N[a]), Sum(Physics:-`*`(Physics:-Ket(B, i), Physics:-Bra(B, i)), i = 1 .. N[b]))

(3)

proj.Ket(Psi)

Sum(Sum(C[i, j]*Physics:-`*`(Physics:-Ket(A, i), Physics:-Ket(B, j)), i = 1 .. N[a]), j = 1 .. N[b])

(4)

NULL

proj2.Ket(Psi)

Sum(Sum(Physics:-`*`(Physics:-Ket(A, i__1), Physics:-Bra(A, i__1), Physics:-Ket(alpha, i), Physics:-Ket(B, i)), i = 1 .. N[b]), i__1 = 1 .. N[a])

(5)

proj-proj2

Sum(Sum(Physics:-`*`(Physics:-Ket(A, i), Physics:-Ket(B, j), Physics:-Bra(A, i), Physics:-Bra(B, j)), i = 1 .. N[a]), j = 1 .. N[b])-Physics:-`*`(Sum(Physics:-`*`(Physics:-Ket(A, i), Physics:-Bra(A, i)), i = 1 .. N[a]), Sum(Physics:-`*`(Physics:-Ket(B, i), Physics:-Bra(B, i)), i = 1 .. N[b]))

(6)

``


 

Download projector_2d_space.mw

 

Two lines that look the same, produce different results. The first lines gives an error message, but the next line that looks the same, does not.

copying and pasting both lines in Notepad reveals the difference:

Determinant*(R1 . B+R2 . B+R3 . B+R4 . A)

Determinant(R1 . B+R2 . B+R3 . B+R4 . A)

It seems that there is a hidden character (the asterisk) in the first line that produces the error.

In the worksheet itself you cannot see the asterisk, but using the arrow keys you can notice that there is another character.

It's hard to debug your code if there are hidden characters.

``

restart; with(LinearAlgebra)

kernelopts(version)*interface(version)

`Maple 2018.2, X86 64 WINDOWS, Nov 16 2018, Build ID 1362973`*`Standard Worksheet Interface, Maple 2018.2, Windows 10, November 16 2018 Build ID 1362973`

(1)

A := Matrix(4, 4, symbol = a, shape = symmetric)

B := Matrix(4, 4, symbol = b, shape = symmetric)

R1 := Matrix(4, 4); R1[1, 1] := 1; R2 := Matrix(4, 4); R2[2, 2] := 1; R3 := Matrix(4, 4); R3[3, 3] := 1; R4 := Matrix(4, 4); R4[4, 4] := 1

Determinant*(R1.B+R2.B+R3.B+R4.A)

Error, (in LinearAlgebra:-Multiply) invalid arguments

 

Determinant(R1.B+R2.B+R3.B+R4.A)

-a[1, 4]*b[1, 2]*b[2, 3]*b[3, 4]+a[1, 4]*b[1, 2]*b[2, 4]*b[3, 3]+a[1, 4]*b[1, 3]*b[2, 2]*b[3, 4]-a[1, 4]*b[1, 3]*b[2, 3]*b[2, 4]-a[1, 4]*b[1, 4]*b[2, 2]*b[3, 3]+a[1, 4]*b[1, 4]*b[2, 3]^2+a[2, 4]*b[1, 1]*b[2, 3]*b[3, 4]-a[2, 4]*b[1, 1]*b[2, 4]*b[3, 3]-a[2, 4]*b[1, 2]*b[1, 3]*b[3, 4]+a[2, 4]*b[1, 2]*b[1, 4]*b[3, 3]+a[2, 4]*b[1, 3]^2*b[2, 4]-a[2, 4]*b[1, 3]*b[1, 4]*b[2, 3]-a[3, 4]*b[1, 1]*b[2, 2]*b[3, 4]+a[3, 4]*b[1, 1]*b[2, 3]*b[2, 4]+a[3, 4]*b[1, 2]^2*b[3, 4]-a[3, 4]*b[1, 2]*b[1, 3]*b[2, 4]-a[3, 4]*b[1, 2]*b[1, 4]*b[2, 3]+a[3, 4]*b[1, 3]*b[1, 4]*b[2, 2]+a[4, 4]*b[1, 1]*b[2, 2]*b[3, 3]-a[4, 4]*b[1, 1]*b[2, 3]^2-a[4, 4]*b[1, 2]^2*b[3, 3]+2*a[4, 4]*b[1, 2]*b[1, 3]*b[2, 3]-a[4, 4]*b[1, 3]^2*b[2, 2]

(2)

``


 

Download weird.mw

i want to generate grid for a C_D nozzle but i dont know ho to plot it. please help me how to plot.

restart;
alpha := 0;
beta := 2;
Imax := 11;
Jmax := 11;
L := 10;
`Δζ` := L/(Imax-1);

`Δx` := `Δζ`;
`Δη` := 1/(Jmax-1);

printlevel := 2;
for i to Imax do for j to Jmax do x[i] := `Δζ`*(i-1); eta[j] := `Δη`*(j-1); z := (eta[j]-alpha)*ln((beta+1)/(beta-1))/(1-alpha); H[i] := 3+2*tanh(x[i]-6); H1[i] := 0; H2[i] := -H[i]; y[i, j] := -H[i]*(2*beta/(exp(z)+1)-beta-2*alpha)/(2*alpha+1); Y[i, j] := (H1[i]+(-H1[i]*x[i]+H2[i])/L)*eta[j] end do end do;

 

 

If I have an expression like this

f:=ln((1-x)^2*(x+1)^2/((-I*x-I+sqrt(-x^2+1))^2*(I*x+I+sqrt(-x^2+1))^2))

maple has trouble to simplify the argument.

In particular is it possible to apply expand() only to the denominator?

This is meant in general, so if I have many terms with expressions like this (possibly of products with other functions in each term), I want this simplification to be done termwise for the arguments of the functions.

Expanding the fraction doesn't work as in frontend(expand, [f]).

Hi,

I am looking to test ANOVA ( Two Way).Maple does not directly gives this possibility.I found a procedure from Mr Tannis but i can t run it? Some ideas?

ThanksANOVA2.mw

This is not a problem per se, but more to understand the background.

restart;

f := polylog(2, -x);

int(f/(x+1), x);

convert(f, dilog);

int(%/(x+1), x)

 

The integration of the polylog maple is not capable of doing, but after converting to dilog it finds an anti derivative.

That leads to the question, why is dilog as a separate to polylog(2,*) implemented anyway? Why couldn't it all be done with the more general polylog function?

 

I'm also wondering why maple has difficulties to integrate

int(dilog(x+1)/(x+a),x)

for general a.

How would I go about getting true or false returned on these propositions?
I have tried just about every eval and various syntax methods, but nothing has worked so far.

I know most can easilly be done by hand/thinking, but I'm sure Maple should have a way to do this as well.

∀n∈Z:2n>n+2   ,   ∃n∈Z:2|(3n+1)    ,   ∃k∈Z:∀n∈Z:n=kn   ,   ∃k∈Z:∀n∈Z:2|(n+k)   ,   ∀n∈Z:∀k∈Z:(n>k∨k≥n)

Using Maple 2018.2.1, I'm receiving a lost kernel message when importing the attached data file with ImportMatrix. I traced the issue to a "*" symbol at the end of the file but would have expected this to cause an error message (if any error at all) instead of the connection to the kernel to be lost. Is this a bug or am I misunderstanding the usage of ImportMatrix?

test.mw

test2.txt

Hi 
How we can obtain inversion of the high-dimensional matrix (for example 600*600) in maple?
(A fast-time command)
thanks


i want someone hlep me in this worksheet the diff eq of complex i want to sovle it with any numeric method 
 

restart

with(Physics):

with(IntegrationTools):

v := 1;

1

 

-500

 

.1

 

.5

 

.5

(1)

``

M[1] := Int(-Physics:-`*`(Physics:-`*`(Physics:-`*`(2, I), exp(-Physics:-`*`(Physics:-`*`(Physics:-`*`(.5, I), v), tt))), 1/Physics:-`*`(Physics:-`*`(sqrt(b^2+tt^2), v^2), exp(sqrt(b^2+tt^2))))+Physics:-`*`(Physics:-`*`(Physics:-`*`(2, I), exp(Physics:-`*`(Physics:-`*`(Physics:-`*`(.5, I), v), tt))), 1/Physics:-`*`(Physics:-`*`(sqrt(b^2+tt^2), v^2), exp(sqrt(b^2+tt^2)))), tt = -500 .. z)

Int(-(2*I)*exp(-(.5*I)*tt)/((0.1e-1+tt^2)^(1/2)*exp((0.1e-1+tt^2)^(1/2)))+(2*I)*exp((.5*I)*tt)/((0.1e-1+tt^2)^(1/2)*exp((0.1e-1+tt^2)^(1/2))), tt = -500 .. z)

(2)

M[2] := Int(Physics:-`*`(Physics:-`*`(Physics:-`*`(Physics:-`*`(4, I), tt), exp(-Physics:-`*`(Physics:-`*`(Physics:-`*`(.5, I), v), tt))), 1/Physics:-`*`(Physics:-`*`(sqrt(b^2+tt^2), v^2), exp(sqrt(b^2+tt^2))))-Physics:-`*`(Physics:-`*`(Physics:-`*`(Physics:-`*`(4, I), tt), exp(Physics:-`*`(Physics:-`*`(Physics:-`*`(.5, I), v), tt))), 1/Physics:-`*`(Physics:-`*`(sqrt(b^2+tt^2), v^2), exp(sqrt(b^2+tt^2)))), tt = -500 .. z):

M[3] := Int(-Physics:-`*`(Physics:-`*`(Physics:-`*`(Physics:-`*`(2, I), tt^2), exp(-Physics:-`*`(Physics:-`*`(Physics:-`*`(.5, I), v), tt))), 1/Physics:-`*`(Physics:-`*`(sqrt(b^2+tt^2), v^2), exp(sqrt(b^2+tt^2))))+Physics:-`*`(Physics:-`*`(Physics:-`*`(Physics:-`*`(2, I), tt^2), exp(Physics:-`*`(Physics:-`*`(Physics:-`*`(.5, I), v), tt))), 1/Physics:-`*`(Physics:-`*`(sqrt(b^2+tt^2), v^2), exp(sqrt(b^2+tt^2)))), tt = -500 .. z):

M[4] := Int(-Physics:-`*`(Physics:-`*`(2, exp(-Physics:-`*`(Physics:-`*`(Physics:-`*`(.5, I), v), tt))), 1/Physics:-`*`(v, exp(sqrt(b^2+tt^2))))-Physics:-`*`(Physics:-`*`(2, exp(Physics:-`*`(Physics:-`*`(Physics:-`*`(.5, I), v), tt))), 1/Physics:-`*`(v, exp(sqrt(b^2+tt^2))))+Physics:-`*`(Physics:-`*`(Physics:-`*`(2, tt), exp(-Physics:-`*`(Physics:-`*`(Physics:-`*`(.5, I), v), tt))), 1/Physics:-`*`(v, exp(sqrt(b^2+tt^2))))+Physics:-`*`(Physics:-`*`(Physics:-`*`(2, tt), exp(Physics:-`*`(Physics:-`*`(Physics:-`*`(.5, I), v), tt))), 1/Physics:-`*`(v, exp(sqrt(b^2+tt^2)))), tt = -500 .. z):

M := Physics:-`*`(z^2, M[1])+Physics:-`*`(z, M[2])+Physics:-`*`(z, M[3])+M[4]:

Mc[1] := Physics:-`*`(z^2, Int(-Physics:-`*`(Physics:-`*`(Physics:-`*`(2, I), exp(-Physics:-`*`(Physics:-`*`(Physics:-`*`(.5, I), v), tt))), 1/Physics:-`*`(Physics:-`*`(sqrt(b^2+tt^2), v^2), exp(sqrt(b^2+tt^2))))+Physics:-`*`(Physics:-`*`(Physics:-`*`(2, I), exp(Physics:-`*`(Physics:-`*`(Physics:-`*`(.5, I), v), tt))), 1/Physics:-`*`(Physics:-`*`(sqrt(b^2+tt^2), v^2), exp(sqrt(b^2+tt^2)))), tt = -500 .. z)):

Mc[2] := Physics:-`*`(z, Int(Physics:-`*`(Physics:-`*`(Physics:-`*`(Physics:-`*`(4, I), tt), exp(-Physics:-`*`(Physics:-`*`(Physics:-`*`(.5, I), v), tt))), 1/Physics:-`*`(Physics:-`*`(sqrt(b^2+tt^2), v^2), exp(sqrt(b^2+tt^2))))-Physics:-`*`(Physics:-`*`(Physics:-`*`(Physics:-`*`(4, I), tt), exp(Physics:-`*`(Physics:-`*`(Physics:-`*`(.5, I), v), tt))), 1/Physics:-`*`(Physics:-`*`(sqrt(b^2+tt^2), v^2), exp(sqrt(b^2+tt^2)))), tt = -500 .. z)):

Mc[3] := Physics:-`*`(z, Int(-Physics:-`*`(Physics:-`*`(2, exp(-Physics:-`*`(Physics:-`*`(Physics:-`*`(.5, I), v), tt))), 1/Physics:-`*`(v, exp(sqrt(b^2+tt^2))))-Physics:-`*`(Physics:-`*`(2, exp(Physics:-`*`(Physics:-`*`(Physics:-`*`(.5, I), v), tt))), 1/Physics:-`*`(v, exp(sqrt(b^2+tt^2)))), tt = -500 .. z)):

Mc[4] := Int(-Physics:-`*`(Physics:-`*`(Physics:-`*`(Physics:-`*`(2, I), tt^2), exp(-Physics:-`*`(Physics:-`*`(Physics:-`*`(.5, I), v), tt))), 1/Physics:-`*`(Physics:-`*`(sqrt(b^2+tt^2), v^2), exp(sqrt(b^2+tt^2))))+Physics:-`*`(Physics:-`*`(Physics:-`*`(Physics:-`*`(2, I), tt^2), exp(Physics:-`*`(Physics:-`*`(Physics:-`*`(.5, I), v), tt))), 1/Physics:-`*`(Physics:-`*`(sqrt(b^2+tt^2), v^2), exp(sqrt(b^2+tt^2))))+Physics:-`*`(Physics:-`*`(Physics:-`*`(2, tt), exp(-Physics:-`*`(Physics:-`*`(Physics:-`*`(.5, I), v), tt))), 1/Physics:-`*`(v, exp(sqrt(b^2+tt^2))))+Physics:-`*`(Physics:-`*`(Physics:-`*`(2, tt), exp(Physics:-`*`(Physics:-`*`(Physics:-`*`(.5, I), v), tt))), 1/Physics:-`*`(v, exp(sqrt(b^2+tt^2)))), tt = -500 .. z):

Mc := Mc[1]+Mc[2]+Mc[3]+Mc[4]:

N[1] := Int(Physics:-`*`(Physics:-`*`(Physics:-`*`(2, tt), exp(-Physics:-`*`(Physics:-`*`(Physics:-`*`(.5, I), v), tt))), 1/Physics:-`*`(Physics:-`*`(sqrt(b^2+tt^2), v), exp(sqrt(b^2+tt^2))))-Physics:-`*`(Physics:-`*`(Physics:-`*`(2, tt), exp(Physics:-`*`(Physics:-`*`(Physics:-`*`(.5, I), v), tt))), 1/Physics:-`*`(Physics:-`*`(sqrt(b^2+tt^2), v), exp(sqrt(b^2+tt^2))))+Physics:-`*`(Physics:-`*`(Physics:-`*`(2, I), exp(-Physics:-`*`(Physics:-`*`(Physics:-`*`(.5, I), v), tt))), 1/exp(sqrt(b^2+tt^2))), tt = -500 .. z):

N[2] := Physics:-`*`(z, Int(-Physics:-`*`(Physics:-`*`(2, exp(-Physics:-`*`(Physics:-`*`(Physics:-`*`(.5, I), v), tt))), 1/Physics:-`*`(Physics:-`*`(sqrt(b^2+tt^2), v), exp(sqrt(b^2+tt^2))))+Physics:-`*`(Physics:-`*`(2, exp(Physics:-`*`(Physics:-`*`(Physics:-`*`(.5, I), v), tt))), 1/Physics:-`*`(Physics:-`*`(sqrt(b^2+tt^2), v), exp(sqrt(b^2+tt^2)))), tt = -500 .. z)):

N := N[1]+N[2]:

Nc[1] := Int(-Physics:-`*`(Physics:-`*`(Physics:-`*`(2., tt), exp(Physics:-`*`(Physics:-`*`(Physics:-`*`(.5, I), v), tt))), 1/Physics:-`*`(Physics:-`*`(sqrt(b^2+tt^2), v), exp(sqrt(b^2+tt^2))))+Physics:-`*`(Physics:-`*`(Physics:-`*`(2., tt), exp(-Physics:-`*`(Physics:-`*`(Physics:-`*`(.5, I), v), tt))), 1/Physics:-`*`(Physics:-`*`(sqrt(b^2+tt^2), v), exp(sqrt(b^2+tt^2))))-Physics:-`*`(Physics:-`*`(Physics:-`*`(2., I), exp(-Physics:-`*`(Physics:-`*`(Physics:-`*`(.5, I), v), tt))), 1/exp(sqrt(b^2+tt^2))), tt = -500 .. z):

Nc[2] := Physics:-`*`(z, Int(Physics:-`*`(Physics:-`*`(2., exp(Physics:-`*`(Physics:-`*`(Physics:-`*`(.5, I), v), tt))), 1/Physics:-`*`(Physics:-`*`(sqrt(b^2+tt^2), v), exp(sqrt(b^2+tt^2))))-Physics:-`*`(Physics:-`*`(2., exp(-Physics:-`*`(Physics:-`*`(Physics:-`*`(.5, I), v), tt))), 1/Physics:-`*`(Physics:-`*`(sqrt(b^2+tt^2), v), exp(sqrt(b^2+tt^2)))), tt = -500 .. z)):

Nc := Nc[1]+Nc[2]:

V := Physics:-`*`(Physics:-`*`(1/Physics:-`*`(4, Pi^2), 1/sqrt(b^2+z^2)), Physics:-`*`(exp(-Physics:-`*`(2, sqrt(b^2+z^2))), Physics:-`*`(2, sqrt(b^2+z^2))+2)-2):

Vc := Physics:-`*`(Physics:-`*`(Physics:-`*`(-1, 1/Physics:-`*`(4, Pi^2)), 1/sqrt(b^2+z^2)), Physics:-`*`(exp(-Physics:-`*`(2, sqrt(b^2+z^2))), Physics:-`*`(2, sqrt(b^2+z^2))+2)-2):

``

H := proc (z) local t; if not z::numeric then return ('procname')(args) end if; evalf(-I*(z^2*(Int(-(2*I)*exp((-1)*.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2)))+(2*I)*exp(.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2))), tt = -500 .. z))+z*(Int((4*I)*tt*exp((-1)*.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2)))-(4*I)*tt*exp(.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2))), tt = -500 .. z))+z*(Int(-2*exp((-1)*.5*I*v*tt)/(v*exp(sqrt(b^2+tt^2)))-2*exp(.5*I*v*tt)/(v*exp(sqrt(b^2+tt^2))), tt = -500 .. z))+Int(-(2*I)*tt^2*exp((-1)*.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2)))+(2*I)*tt^2*exp(.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2)))+2*tt*exp((-1)*.5*I*v*tt)/(v*exp(sqrt(b^2+tt^2)))+2*tt*exp(.5*I*v*tt)/(v*exp(sqrt(b^2+tt^2))), tt = -500 .. z))*(-1/2+(1/2)*p^2*v^2+1/sqrt(b^2+z^2)+(1/4)*(exp(-2*sqrt(b^2+z^2))*(2*sqrt(b^2+z^2)+2)-2)/(Pi^2*sqrt(b^2+z^2)))/(((z^2*(Int(-(2*I)*exp((-1)*.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2)))+(2*I)*exp(.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2))), tt = -500 .. z))+z*(Int((4*I)*tt*exp((-1)*.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2)))-(4*I)*tt*exp(.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2))), tt = -500 .. z))+z*(Int(-(2*I)*tt^2*exp((-1)*.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2)))+(2*I)*tt^2*exp(.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2))), tt = -500 .. z))+Int(-2*exp((-1)*.5*I*v*tt)/(v*exp(sqrt(b^2+tt^2)))-2*exp(.5*I*v*tt)/(v*exp(sqrt(b^2+tt^2)))+2*tt*exp((-1)*.5*I*v*tt)/(v*exp(sqrt(b^2+tt^2)))+2*tt*exp(.5*I*v*tt)/(v*exp(sqrt(b^2+tt^2))), tt = -500 .. z))*(z^2*(Int(-(2*I)*exp((-1)*.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2)))+(2*I)*exp(.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2))), tt = -500 .. z))+z*(Int((4*I)*tt*exp((-1)*.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2)))-(4*I)*tt*exp(.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2))), tt = -500 .. z))+z*(Int(-2*exp((-1)*.5*I*v*tt)/(v*exp(sqrt(b^2+tt^2)))-2*exp(.5*I*v*tt)/(v*exp(sqrt(b^2+tt^2))), tt = -500 .. z))+Int(-(2*I)*tt^2*exp((-1)*.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2)))+(2*I)*tt^2*exp(.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2)))+2*tt*exp((-1)*.5*I*v*tt)/(v*exp(sqrt(b^2+tt^2)))+2*tt*exp(.5*I*v*tt)/(v*exp(sqrt(b^2+tt^2))), tt = -500 .. z))-1)*v)+I*((-1/2+(1/2)*p^2*v^2+1/sqrt(b^2+z^2))*(z^2*(Int(-(2*I)*exp((-1)*.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2)))+(2*I)*exp(.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2))), tt = -500 .. z))+z*(Int((4*I)*tt*exp((-1)*.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2)))-(4*I)*tt*exp(.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2))), tt = -500 .. z))+z*(Int(-2*exp((-1)*.5*I*v*tt)/(v*exp(sqrt(b^2+tt^2)))-2*exp(.5*I*v*tt)/(v*exp(sqrt(b^2+tt^2))), tt = -500 .. z))+Int(-(2*I)*tt^2*exp((-1)*.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2)))+(2*I)*tt^2*exp(.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2)))+2*tt*exp((-1)*.5*I*v*tt)/(v*exp(sqrt(b^2+tt^2)))+2*tt*exp(.5*I*v*tt)/(v*exp(sqrt(b^2+tt^2))), tt = -500 .. z))+Int((-1)*2.*tt*exp(.5*I*v*tt)/(sqrt(b^2+tt^2)*v*exp(sqrt(b^2+tt^2)))+2.*tt*exp((-1)*.5*I*v*tt)/(sqrt(b^2+tt^2)*v*exp(sqrt(b^2+tt^2)))+(-1)*2.*I*exp((-1)*.5*I*v*tt)/exp(sqrt(b^2+tt^2)), tt = -500 .. z)+z*(Int(2.*exp(.5*I*v*tt)/(sqrt(b^2+tt^2)*v*exp(sqrt(b^2+tt^2)))+(-1)*2.*exp((-1)*.5*I*v*tt)/(sqrt(b^2+tt^2)*v*exp(sqrt(b^2+tt^2))), tt = -500 .. z)))/(((z^2*(Int(-(2*I)*exp((-1)*.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2)))+(2*I)*exp(.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2))), tt = -500 .. z))+z*(Int((4*I)*tt*exp((-1)*.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2)))-(4*I)*tt*exp(.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2))), tt = -500 .. z))+z*(Int(-(2*I)*tt^2*exp((-1)*.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2)))+(2*I)*tt^2*exp(.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2))), tt = -500 .. z))+Int(-2*exp((-1)*.5*I*v*tt)/(v*exp(sqrt(b^2+tt^2)))-2*exp(.5*I*v*tt)/(v*exp(sqrt(b^2+tt^2)))+2*tt*exp((-1)*.5*I*v*tt)/(v*exp(sqrt(b^2+tt^2)))+2*tt*exp(.5*I*v*tt)/(v*exp(sqrt(b^2+tt^2))), tt = -500 .. z))*(z^2*(Int(-(2*I)*exp((-1)*.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2)))+(2*I)*exp(.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2))), tt = -500 .. z))+z*(Int((4*I)*tt*exp((-1)*.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2)))-(4*I)*tt*exp(.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2))), tt = -500 .. z))+z*(Int(-2*exp((-1)*.5*I*v*tt)/(v*exp(sqrt(b^2+tt^2)))-2*exp(.5*I*v*tt)/(v*exp(sqrt(b^2+tt^2))), tt = -500 .. z))+Int(-(2*I)*tt^2*exp((-1)*.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2)))+(2*I)*tt^2*exp(.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2)))+2*tt*exp((-1)*.5*I*v*tt)/(v*exp(sqrt(b^2+tt^2)))+2*tt*exp(.5*I*v*tt)/(v*exp(sqrt(b^2+tt^2))), tt = -500 .. z))-1)*v)) end proc:

H(500)

-6.287499768+0.1713975e-19*I

(3)

NULL

L := proc (z) local t; if not z::numeric then return ('procname')(args) end if; evalf(-I*(z^2*(Int(-(2*I)*exp((-1)*.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2)))+(2*I)*exp(.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2))), tt = -500 .. z))+z*(Int((4*I)*tt*exp((-1)*.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2)))-(4*I)*tt*exp(.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2))), tt = -500 .. z))+z*(Int(-2*exp((-1)*.5*I*v*tt)/(v*exp(sqrt(b^2+tt^2)))-2*exp(.5*I*v*tt)/(v*exp(sqrt(b^2+tt^2))), tt = -500 .. z))+Int(-(2*I)*tt^2*exp((-1)*.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2)))+(2*I)*tt^2*exp(.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2)))+2*tt*exp((-1)*.5*I*v*tt)/(v*exp(sqrt(b^2+tt^2)))+2*tt*exp(.5*I*v*tt)/(v*exp(sqrt(b^2+tt^2))), tt = -500 .. z))*((-1/2+(1/2)*q^2*v^2+1/sqrt(b^2+z^2))*(z^2*(Int(-(2*I)*exp((-1)*.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2)))+(2*I)*exp(.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2))), tt = -500 .. z))+z*(Int((4*I)*tt*exp((-1)*.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2)))-(4*I)*tt*exp(.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2))), tt = -500 .. z))+z*(Int(-(2*I)*tt^2*exp((-1)*.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2)))+(2*I)*tt^2*exp(.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2))), tt = -500 .. z))+Int(-2*exp((-1)*.5*I*v*tt)/(v*exp(sqrt(b^2+tt^2)))-2*exp(.5*I*v*tt)/(v*exp(sqrt(b^2+tt^2)))+2*tt*exp((-1)*.5*I*v*tt)/(v*exp(sqrt(b^2+tt^2)))+2*tt*exp(.5*I*v*tt)/(v*exp(sqrt(b^2+tt^2))), tt = -500 .. z))+Int(2*tt*exp((-1)*.5*I*v*tt)/(sqrt(b^2+tt^2)*v*exp(sqrt(b^2+tt^2)))-2*tt*exp(.5*I*v*tt)/(sqrt(b^2+tt^2)*v*exp(sqrt(b^2+tt^2)))+(2*I)*exp((-1)*.5*I*v*tt)/exp(sqrt(b^2+tt^2)), tt = -500 .. z)+z*(Int(-2*exp((-1)*.5*I*v*tt)/(sqrt(b^2+tt^2)*v*exp(sqrt(b^2+tt^2)))+2*exp(.5*I*v*tt)/(sqrt(b^2+tt^2)*v*exp(sqrt(b^2+tt^2))), tt = -500 .. z)))/(((z^2*(Int(-(2*I)*exp((-1)*.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2)))+(2*I)*exp(.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2))), tt = -500 .. z))+z*(Int((4*I)*tt*exp((-1)*.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2)))-(4*I)*tt*exp(.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2))), tt = -500 .. z))+z*(Int(-(2*I)*tt^2*exp((-1)*.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2)))+(2*I)*tt^2*exp(.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2))), tt = -500 .. z))+Int(-2*exp((-1)*.5*I*v*tt)/(v*exp(sqrt(b^2+tt^2)))-2*exp(.5*I*v*tt)/(v*exp(sqrt(b^2+tt^2)))+2*tt*exp((-1)*.5*I*v*tt)/(v*exp(sqrt(b^2+tt^2)))+2*tt*exp(.5*I*v*tt)/(v*exp(sqrt(b^2+tt^2))), tt = -500 .. z))*(z^2*(Int(-(2*I)*exp((-1)*.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2)))+(2*I)*exp(.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2))), tt = -500 .. z))+z*(Int((4*I)*tt*exp((-1)*.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2)))-(4*I)*tt*exp(.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2))), tt = -500 .. z))+z*(Int(-2*exp((-1)*.5*I*v*tt)/(v*exp(sqrt(b^2+tt^2)))-2*exp(.5*I*v*tt)/(v*exp(sqrt(b^2+tt^2))), tt = -500 .. z))+Int(-(2*I)*tt^2*exp((-1)*.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2)))+(2*I)*tt^2*exp(.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2)))+2*tt*exp((-1)*.5*I*v*tt)/(v*exp(sqrt(b^2+tt^2)))+2*tt*exp(.5*I*v*tt)/(v*exp(sqrt(b^2+tt^2))), tt = -500 .. z))-1)*v)+I*(-1/2+(1/2)*q^2*v^2+1/sqrt(b^2+z^2)-(1/4)*(exp(-2*sqrt(b^2+z^2))*(2*sqrt(b^2+z^2)+2)-2)/(Pi^2*sqrt(b^2+z^2)))/(((z^2*(Int(-(2*I)*exp((-1)*.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2)))+(2*I)*exp(.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2))), tt = -500 .. z))+z*(Int((4*I)*tt*exp((-1)*.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2)))-(4*I)*tt*exp(.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2))), tt = -500 .. z))+z*(Int(-(2*I)*tt^2*exp((-1)*.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2)))+(2*I)*tt^2*exp(.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2))), tt = -500 .. z))+Int(-2*exp((-1)*.5*I*v*tt)/(v*exp(sqrt(b^2+tt^2)))-2*exp(.5*I*v*tt)/(v*exp(sqrt(b^2+tt^2)))+2*tt*exp((-1)*.5*I*v*tt)/(v*exp(sqrt(b^2+tt^2)))+2*tt*exp(.5*I*v*tt)/(v*exp(sqrt(b^2+tt^2))), tt = -500 .. z))*(z^2*(Int(-(2*I)*exp((-1)*.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2)))+(2*I)*exp(.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2))), tt = -500 .. z))+z*(Int((4*I)*tt*exp((-1)*.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2)))-(4*I)*tt*exp(.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2))), tt = -500 .. z))+z*(Int(-2*exp((-1)*.5*I*v*tt)/(v*exp(sqrt(b^2+tt^2)))-2*exp(.5*I*v*tt)/(v*exp(sqrt(b^2+tt^2))), tt = -500 .. z))+Int(-(2*I)*tt^2*exp((-1)*.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2)))+(2*I)*tt^2*exp(.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2)))+2*tt*exp((-1)*.5*I*v*tt)/(v*exp(sqrt(b^2+tt^2)))+2*tt*exp(.5*I*v*tt)/(v*exp(sqrt(b^2+tt^2))), tt = -500 .. z))-1)*v)) end proc:

``

G := proc (z) local t; if not z::numeric then return ('procname')(args) end if; evalf(I*(-1/2+(1/2)*p^2*v^2+1/sqrt(b^2+z^2)+(1/4)*(exp(-2*sqrt(b^2+z^2))*(2*sqrt(b^2+z^2)+2)-2)/(Pi^2*sqrt(b^2+z^2)))/(((z^2*(Int(-(2*I)*exp((-1)*.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2)))+(2*I)*exp(.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2))), tt = -500 .. z))+z*(Int((4*I)*tt*exp((-1)*.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2)))-(4*I)*tt*exp(.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2))), tt = -500 .. z))+z*(Int(-(2*I)*tt^2*exp((-1)*.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2)))+(2*I)*tt^2*exp(.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2))), tt = -500 .. z))+Int(-2*exp((-1)*.5*I*v*tt)/(v*exp(sqrt(b^2+tt^2)))-2*exp(.5*I*v*tt)/(v*exp(sqrt(b^2+tt^2)))+2*tt*exp((-1)*.5*I*v*tt)/(v*exp(sqrt(b^2+tt^2)))+2*tt*exp(.5*I*v*tt)/(v*exp(sqrt(b^2+tt^2))), tt = -500 .. z))*(z^2*(Int(-(2*I)*exp((-1)*.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2)))+(2*I)*exp(.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2))), tt = -500 .. z))+z*(Int((4*I)*tt*exp((-1)*.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2)))-(4*I)*tt*exp(.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2))), tt = -500 .. z))+z*(Int(-2*exp((-1)*.5*I*v*tt)/(v*exp(sqrt(b^2+tt^2)))-2*exp(.5*I*v*tt)/(v*exp(sqrt(b^2+tt^2))), tt = -500 .. z))+Int(-(2*I)*tt^2*exp((-1)*.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2)))+(2*I)*tt^2*exp(.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2)))+2*tt*exp((-1)*.5*I*v*tt)/(v*exp(sqrt(b^2+tt^2)))+2*tt*exp(.5*I*v*tt)/(v*exp(sqrt(b^2+tt^2))), tt = -500 .. z))-1)*v)-I*(z^2*(Int(-(2*I)*exp((-1)*.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2)))+(2*I)*exp(.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2))), tt = -500 .. z))+z*(Int((4*I)*tt*exp((-1)*.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2)))-(4*I)*tt*exp(.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2))), tt = -500 .. z))+z*(Int(-(2*I)*tt^2*exp((-1)*.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2)))+(2*I)*tt^2*exp(.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2))), tt = -500 .. z))+Int(-2*exp((-1)*.5*I*v*tt)/(v*exp(sqrt(b^2+tt^2)))-2*exp(.5*I*v*tt)/(v*exp(sqrt(b^2+tt^2)))+2*tt*exp((-1)*.5*I*v*tt)/(v*exp(sqrt(b^2+tt^2)))+2*tt*exp(.5*I*v*tt)/(v*exp(sqrt(b^2+tt^2))), tt = -500 .. z))*((-1/2+(1/2)*p^2*v^2+1/sqrt(b^2+z^2))*(z^2*(Int(-(2*I)*exp((-1)*.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2)))+(2*I)*exp(.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2))), tt = -500 .. z))+z*(Int((4*I)*tt*exp((-1)*.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2)))-(4*I)*tt*exp(.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2))), tt = -500 .. z))+z*(Int(-2*exp((-1)*.5*I*v*tt)/(v*exp(sqrt(b^2+tt^2)))-2*exp(.5*I*v*tt)/(v*exp(sqrt(b^2+tt^2))), tt = -500 .. z))+Int(-(2*I)*tt^2*exp((-1)*.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2)))+(2*I)*tt^2*exp(.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2)))+2*tt*exp((-1)*.5*I*v*tt)/(v*exp(sqrt(b^2+tt^2)))+2*tt*exp(.5*I*v*tt)/(v*exp(sqrt(b^2+tt^2))), tt = -500 .. z))+Int((-1)*2.*tt*exp(.5*I*v*tt)/(sqrt(b^2+tt^2)*v*exp(sqrt(b^2+tt^2)))+2.*tt*exp((-1)*.5*I*v*tt)/(sqrt(b^2+tt^2)*v*exp(sqrt(b^2+tt^2)))+(-1)*2.*I*exp((-1)*.5*I*v*tt)/exp(sqrt(b^2+tt^2)), tt = -500 .. z)+z*(Int(2.*exp(.5*I*v*tt)/(sqrt(b^2+tt^2)*v*exp(sqrt(b^2+tt^2)))+(-1)*2.*exp((-1)*.5*I*v*tt)/(sqrt(b^2+tt^2)*v*exp(sqrt(b^2+tt^2))), tt = -500 .. z)))/(((z^2*(Int(-(2*I)*exp((-1)*.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2)))+(2*I)*exp(.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2))), tt = -500 .. z))+z*(Int((4*I)*tt*exp((-1)*.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2)))-(4*I)*tt*exp(.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2))), tt = -500 .. z))+z*(Int(-(2*I)*tt^2*exp((-1)*.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2)))+(2*I)*tt^2*exp(.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2))), tt = -500 .. z))+Int(-2*exp((-1)*.5*I*v*tt)/(v*exp(sqrt(b^2+tt^2)))-2*exp(.5*I*v*tt)/(v*exp(sqrt(b^2+tt^2)))+2*tt*exp((-1)*.5*I*v*tt)/(v*exp(sqrt(b^2+tt^2)))+2*tt*exp(.5*I*v*tt)/(v*exp(sqrt(b^2+tt^2))), tt = -500 .. z))*(z^2*(Int(-(2*I)*exp((-1)*.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2)))+(2*I)*exp(.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2))), tt = -500 .. z))+z*(Int((4*I)*tt*exp((-1)*.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2)))-(4*I)*tt*exp(.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2))), tt = -500 .. z))+z*(Int(-2*exp((-1)*.5*I*v*tt)/(v*exp(sqrt(b^2+tt^2)))-2*exp(.5*I*v*tt)/(v*exp(sqrt(b^2+tt^2))), tt = -500 .. z))+Int(-(2*I)*tt^2*exp((-1)*.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2)))+(2*I)*tt^2*exp(.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2)))+2*tt*exp((-1)*.5*I*v*tt)/(v*exp(sqrt(b^2+tt^2)))+2*tt*exp(.5*I*v*tt)/(v*exp(sqrt(b^2+tt^2))), tt = -500 .. z))-1)*v)) end proc:

``

K := proc (z) local t; if not z::numeric then return ('procname')(args) end if; evalf(I*((-1/2+(1/2)*q^2*v^2+1/sqrt(b^2+z^2))*(z^2*(Int(-(2*I)*exp((-1)*.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2)))+(2*I)*exp(.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2))), tt = -500 .. z))+z*(Int((4*I)*tt*exp((-1)*.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2)))-(4*I)*tt*exp(.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2))), tt = -500 .. z))+z*(Int(-(2*I)*tt^2*exp((-1)*.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2)))+(2*I)*tt^2*exp(.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2))), tt = -500 .. z))+Int(-2*exp((-1)*.5*I*v*tt)/(v*exp(sqrt(b^2+tt^2)))-2*exp(.5*I*v*tt)/(v*exp(sqrt(b^2+tt^2)))+2*tt*exp((-1)*.5*I*v*tt)/(v*exp(sqrt(b^2+tt^2)))+2*tt*exp(.5*I*v*tt)/(v*exp(sqrt(b^2+tt^2))), tt = -500 .. z))+Int(2*tt*exp((-1)*.5*I*v*tt)/(sqrt(b^2+tt^2)*v*exp(sqrt(b^2+tt^2)))-2*tt*exp(.5*I*v*tt)/(sqrt(b^2+tt^2)*v*exp(sqrt(b^2+tt^2)))+(2*I)*exp((-1)*.5*I*v*tt)/exp(sqrt(b^2+tt^2)), tt = -500 .. z)+z*(Int(-2*exp((-1)*.5*I*v*tt)/(sqrt(b^2+tt^2)*v*exp(sqrt(b^2+tt^2)))+2*exp(.5*I*v*tt)/(sqrt(b^2+tt^2)*v*exp(sqrt(b^2+tt^2))), tt = -500 .. z)))/(((z^2*(Int(-(2*I)*exp((-1)*.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2)))+(2*I)*exp(.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2))), tt = -500 .. z))+z*(Int((4*I)*tt*exp((-1)*.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2)))-(4*I)*tt*exp(.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2))), tt = -500 .. z))+z*(Int(-(2*I)*tt^2*exp((-1)*.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2)))+(2*I)*tt^2*exp(.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2))), tt = -500 .. z))+Int(-2*exp((-1)*.5*I*v*tt)/(v*exp(sqrt(b^2+tt^2)))-2*exp(.5*I*v*tt)/(v*exp(sqrt(b^2+tt^2)))+2*tt*exp((-1)*.5*I*v*tt)/(v*exp(sqrt(b^2+tt^2)))+2*tt*exp(.5*I*v*tt)/(v*exp(sqrt(b^2+tt^2))), tt = -500 .. z))*(z^2*(Int(-(2*I)*exp((-1)*.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2)))+(2*I)*exp(.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2))), tt = -500 .. z))+z*(Int((4*I)*tt*exp((-1)*.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2)))-(4*I)*tt*exp(.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2))), tt = -500 .. z))+z*(Int(-2*exp((-1)*.5*I*v*tt)/(v*exp(sqrt(b^2+tt^2)))-2*exp(.5*I*v*tt)/(v*exp(sqrt(b^2+tt^2))), tt = -500 .. z))+Int(-(2*I)*tt^2*exp((-1)*.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2)))+(2*I)*tt^2*exp(.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2)))+2*tt*exp((-1)*.5*I*v*tt)/(v*exp(sqrt(b^2+tt^2)))+2*tt*exp(.5*I*v*tt)/(v*exp(sqrt(b^2+tt^2))), tt = -500 .. z))-1)*v)-I*(z^2*(Int(-(2*I)*exp((-1)*.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2)))+(2*I)*exp(.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2))), tt = -500 .. z))+z*(Int((4*I)*tt*exp((-1)*.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2)))-(4*I)*tt*exp(.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2))), tt = -500 .. z))+z*(Int(-(2*I)*tt^2*exp((-1)*.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2)))+(2*I)*tt^2*exp(.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2))), tt = -500 .. z))+Int(-2*exp((-1)*.5*I*v*tt)/(v*exp(sqrt(b^2+tt^2)))-2*exp(.5*I*v*tt)/(v*exp(sqrt(b^2+tt^2)))+2*tt*exp((-1)*.5*I*v*tt)/(v*exp(sqrt(b^2+tt^2)))+2*tt*exp(.5*I*v*tt)/(v*exp(sqrt(b^2+tt^2))), tt = -500 .. z))*(-1/2+(1/2)*q^2*v^2+1/sqrt(b^2+z^2)-(1/4)*(exp(-2*sqrt(b^2+z^2))*(2*sqrt(b^2+z^2)+2)-2)/(Pi^2*sqrt(b^2+z^2)))/(((z^2*(Int(-(2*I)*exp((-1)*.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2)))+(2*I)*exp(.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2))), tt = -500 .. z))+z*(Int((4*I)*tt*exp((-1)*.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2)))-(4*I)*tt*exp(.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2))), tt = -500 .. z))+z*(Int(-(2*I)*tt^2*exp((-1)*.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2)))+(2*I)*tt^2*exp(.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2))), tt = -500 .. z))+Int(-2*exp((-1)*.5*I*v*tt)/(v*exp(sqrt(b^2+tt^2)))-2*exp(.5*I*v*tt)/(v*exp(sqrt(b^2+tt^2)))+2*tt*exp((-1)*.5*I*v*tt)/(v*exp(sqrt(b^2+tt^2)))+2*tt*exp(.5*I*v*tt)/(v*exp(sqrt(b^2+tt^2))), tt = -500 .. z))*(z^2*(Int(-(2*I)*exp((-1)*.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2)))+(2*I)*exp(.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2))), tt = -500 .. z))+z*(Int((4*I)*tt*exp((-1)*.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2)))-(4*I)*tt*exp(.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2))), tt = -500 .. z))+z*(Int(-2*exp((-1)*.5*I*v*tt)/(v*exp(sqrt(b^2+tt^2)))-2*exp(.5*I*v*tt)/(v*exp(sqrt(b^2+tt^2))), tt = -500 .. z))+Int(-(2*I)*tt^2*exp((-1)*.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2)))+(2*I)*tt^2*exp(.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2)))+2*tt*exp((-1)*.5*I*v*tt)/(v*exp(sqrt(b^2+tt^2)))+2*tt*exp(.5*I*v*tt)/(v*exp(sqrt(b^2+tt^2))), tt = -500 .. z))-1)*v)) end proc:

NULL

NULL

sys := {diff(X(z), z) = Physics:-`*`(H(z), Y(z))+Physics:-`*`(L(z), X(z)), diff(Y(z), z) = Physics:-`*`(G(z), Y(z))+Physics:-`*`(K(z), X(z))}

{diff(X(z), z) = H(z)*Y(z)+L(z)*X(z), diff(Y(z), z) = G(z)*Y(z)+K(z)*X(z)}

(4)

IC_1 := {X(-500) = 0, Y(-500) = 1}

{X(-500) = 0, Y(-500) = 1}

(5)

dsol3 := dsolve(`union`(sys, IC_1), numeric, method = dverk78, output = procedurelist, known = [H, L, G, K])

proc (x_dverk78) local _res, _dat, _vars, _solnproc, _xout, _ndsol, _pars, _n, _i; option `Copyright (c) 2000 by Waterloo Maple Inc. All rights reserved.`; if 1 < nargs then error "invalid input: too many arguments" end if; _EnvDSNumericSaveDigits := Digits; Digits := 15; if _EnvInFsolve = true then _xout := evalf[_EnvDSNumericSaveDigits](x_dverk78) else _xout := evalf(x_dverk78) end if; _dat := Array(1..4, {(1) = proc (_xin) local _xout, _fcn, _i, _octl, _ctl, _y0, _yini, _ycur, _reinit, _pars, _n, _ysav, _ini, _par; option `Copyright (c) 1993 by the University of Waterloo. All rights reserved.`; Digits := max(15, Digits); _xout := _xin; _octl := array( 1 .. 32, [( 1 ) = (3), ( 2 ) = (1), ( 3 ) = (0), ( 4 ) = (0), ( 5 ) = (0), ( 6 ) = (0), ( 7 ) = (0), ( 9 ) = (1), ( 8 ) = (0), ( 11 ) = (-1), ( 10 ) = (-1), ( 13 ) = (-1), ( 12 ) = (-1), ( 15 ) = (-1), ( 14 ) = (-1), ( 18 ) = (-1), ( 19 ) = (-1), ( 16 ) = (-1), ( 17 ) = (-500.), ( 22 ) = (-1), ( 23 ) = (-1), ( 20 ) = (-500.), ( 21 ) = (-1), ( 27 ) = (-499.), ( 26 ) = (2), ( 25 ) = (-500.), ( 24 ) = (0), ( 31 ) = (-500.), ( 30 ) = (1), ( 29 ) = (2), ( 28 ) = (0.1e-7), ( 32 ) = (0)  ] ); _yini := Array(0..2, {(1) = -500., (2) = 0.}); _y0 := Array(0..2, {(1) = -500., (2) = 0.}); _ycur := array( 1 .. 2, [ ] ); _ctl := array( 1 .. 32, [( 1 ) = (3), ( 2 ) = (1), ( 3 ) = (0), ( 4 ) = (0), ( 5 ) = (0), ( 6 ) = (0), ( 7 ) = (0), ( 9 ) = (1), ( 8 ) = (0), ( 11 ) = (-1), ( 10 ) = (-1), ( 13 ) = (-1), ( 12 ) = (-1), ( 15 ) = (-1), ( 14 ) = (-1), ( 18 ) = (-1), ( 19 ) = (-1), ( 16 ) = (-1), ( 17 ) = (-500.), ( 22 ) = (-1), ( 23 ) = (-1), ( 20 ) = (-500.), ( 21 ) = (-1), ( 27 ) = (-499.), ( 26 ) = (2), ( 25 ) = (-500.), ( 24 ) = (0), ( 31 ) = (-500.), ( 30 ) = (1), ( 29 ) = (2), ( 28 ) = (0.1e-7), ( 32 ) = (0)  ] ); _fcn := proc (N, X, Y, YP) option `[Y[1] = X(z), Y[2] = Y(z)]`; YP[1] := H(X)*Y[2]+L(X)*Y[1]; YP[2] := G(X)*Y[2]+K(X)*Y[1]; 0 end proc; _pars := []; _n := 2; _ysav := Array(1..2, {(1) = 0., (2) = 1.}); if not type(_xout, 'numeric') then if member(_xout, ["start", "left", "right"]) then return _y0[0] elif _xout = "method" then return "dverk78" elif _xout = "numfun" then return round(_ctl[24]) elif _xout = "initial" then return [seq(_yini[_i], _i = 0 .. _n)] elif _xout = "parameters" then return [seq(_yini[_n+_i], _i = 1 .. nops(_pars))] elif _xout = "initial_and_parameters" then return [seq(_yini[_i], _i = 0 .. _n)], [seq(_yini[_n+_i], _i = 1 .. nops(_pars))] elif _xout = "last" then if _ctl[17]-_y0[0] = 0. then error "no information is available on last computed point" else _xout := _ctl[17] end if elif _xout = "enginedata" then return eval(_octl, 1) elif _xout = "function" then return eval(_fcn, 1) elif type(_xin, `=`) and type(rhs(_xin), 'list') and member(lhs(_xin), {"initial", "parameters", "initial_and_parameters"}) then _ini, _par := [], []; if lhs(_xin) = "initial" then _ini := rhs(_xin) elif lhs(_xin) = "parameters" then _par := rhs(_xin) elif select(type, rhs(_xin), `=`) <> [] then _par, _ini := selectremove(type, rhs(_xin), `=`) elif nops(rhs(_xin)) < nops(_pars)+1 then error "insufficient data for specification of initial and parameters" else _par := rhs(_xin)[-nops(_pars) .. -1]; _ini := rhs(_xin)[1 .. -nops(_pars)-1] end if; _xout := lhs(_xout); if _par <> [] then `dsolve/numeric/process_parameters`(_n, _pars, _par, _yini) end if; if _ini <> [] then `dsolve/numeric/process_initial`(_n, _ini, _yini, _pars) end if; if _pars <> [] then _par := {seq(rhs(_pars[_i]) = _yini[_n+_i], _i = 1 .. nops(_par))}; for _i from 0 to _n do _y0[_i] := subs(_par, _yini[_i]) end do; for _i from _n+1 to _n+nops(_pars) do _y0[_i] := _yini[_i] end do else for _i from 0 to _n do _y0[_i] := _yini[_i] end do end if; _octl[25] := _y0[0]; _octl[20] := _y0[0]; _octl[17] := _y0[0]; _octl[31] := _y0[0]; for _i to op(2, op(2, op(_octl))) do _ctl[_i] := _octl[_i] end do; for _i to _n+nops(_pars) do _ysav[_i] := _y0[_i] end do; if _xout = "initial" then return [seq(_yini[_i], _i = 0 .. _n)] elif _xout = "parameters" then return [seq(_yini[_n+_i], _i = 1 .. nops(_pars))] else return [seq(_yini[_i], _i = 0 .. _n)], [seq(_yini[_n+_i], _i = 1 .. nops(_pars))] end if else return "procname" end if end if; if _y0[0]-_xout = 0. then return [seq(_y0[_i], _i = 0 .. _n)] elif _octl[31]-_y0[0] = 0. then _octl[31] := _y0[0]-sign(_xout-_y0[0]) end if; _reinit := false; if _xin <> "last" then if 0 < 0 and `dsolve/numeric/checkglobals`(0, table( [ ] ), _pars, _n, _yini) then _reinit := true; if _pars <> [] then _par := {seq(rhs(_pars[_i]) = _yini[_n+_i], _i = 1 .. nops(_par))}; for _i from 0 to _n do _y0[_i] := subs(_par, _yini[_i]) end do; for _i from _n+1 to _n+nops(_pars) do _y0[_i] := _yini[_i] end do else for _i from 0 to _n do _y0[_i] := _yini[_i] end do end if end if; if _pars <> [] and select(type, {seq(_yini[_n+_i], _i = 1 .. nops(_pars))}, 'undefined') <> {} then error "parameters must be initialized before solution can be computed" end if end if; if _reinit or _ctl[17]-_xout <> 0. then if _reinit or 0 < _ctl[18] and _xout < _ctl[31] or _ctl[18] < 0 and _ctl[31] < _xout then for _i to op(2, op(2, op(_octl))) do _ctl[_i] := _octl[_i] end do; for _i to _n+nops(_pars) do _ysav[_i] := _y0[_i] end do else _ctl[29] := 2 end if; _ctl[9] := 2; _ctl[27] := _xout; if Digits <= trunc(evalhf(Digits)) then try evalhf(`dsolve/numeric/dverk78_engine`(_fcn, var(_ysav), var(_ycur), `dsolve/numeric/dverk78_aa`, `dsolve/numeric/dverk78_cc`, `dsolve/numeric/dverk78_dd`, var(_ctl), var(array( 1 .. 2, [ ] )), var(array( 1 .. 2, [ ] )), var(array( 1 .. 2, 1 .. 23, [ ] )))) catch: if searchtext('evalhf', lastexception[2]) <> 0 or searchtext('real', lastexception[2]) <> 0 or searchtext('hardware', lastexception[2]) <> 0 then `dsolve/numeric/dverk78_engine`(_fcn, _ysav, _ycur, `dsolve/numeric/dverk78_aa`, `dsolve/numeric/dverk78_cc`, `dsolve/numeric/dverk78_dd`, _ctl, array( 1 .. 2, [ ] ), array( 1 .. 2, [ ] ), array( 1 .. 2, 1 .. 23, [ ] )) else error  end if end try else `dsolve/numeric/dverk78_engine`(_fcn, _ysav, _ycur, `dsolve/numeric/dverk78_aa`, `dsolve/numeric/dverk78_cc`, `dsolve/numeric/dverk78_dd`, _ctl, array( 1 .. 2, [ ] ), array( 1 .. 2, [ ] ), array( 1 .. 2, 1 .. 23, [ ] )) end if; if _ctl[29]-3 <> 0 then Rounding := `if`(_y0[0] < _xout, -infinity, infinity); if _ctl[29]+1 = 0 then error "cannot evaluate the solution past %1, maxfun limit exceeded (see <a href='http://www.maplesoft.com/support/help/search.aspx?term=dsolve,maxfun' target='_new'>?dsolve,maxfun</a> for details)", evalf[8](_ctl[20]) elif _ctl[29]+2 = 0 then error "cannot evaluate the solution past %1, hmin > hmax, maybe error tolerance is too small", evalf[8](_ctl[20]) elif _ctl[29]+3 = 0 then error "cannot evaluate the solution past %1, step size < hmin, problem may be singular or error tolerance may be too small", evalf[8](_ctl[20]) else error "cannot evaluate the solution past %1, unknown error code returned from dverk78: %2", evalf[8](_ctl[20]), _ctl[29] end if end if; if _Env_smart_dsolve_numeric = true then if _y0[0] < _ctl[17] and procname("right") < _ctl[17] then procname("right") := _ctl[17] elif _ctl[17] < _y0[0] and _ctl[17] < procname("left") then procname("left") := _ctl[17] end if end if end if; [_ctl[25], seq(_ycur[_i], _i = 1 .. _n)] end proc, (2) = Array(0..0, {}), (3) = [z, X(z), Y(z)], (4) = []}); _vars := _dat[3]; _pars := map(rhs, _dat[4]); _n := nops(_vars)-1; _solnproc := _dat[1]; if not type(_xout, 'numeric') then if member(x_dverk78, ["start", 'start', "method", 'method', "left", 'left', "right", 'right', "leftdata", "rightdata", "enginedata", "eventstop", 'eventstop', "eventclear", 'eventclear', "eventstatus", 'eventstatus', "eventcount", 'eventcount', "laxtol", 'laxtol', "numfun", 'numfun', NULL]) then _res := _solnproc(convert(x_dverk78, 'string')); if 1 < nops([_res]) then return _res elif type(_res, 'array') then return eval(_res, 1) elif _res <> "procname" then return _res end if elif member(x_dverk78, ["last", 'last', "initial", 'initial', "parameters", 'parameters', "initial_and_parameters", 'initial_and_parameters', NULL]) then _xout := convert(x_dverk78, 'string'); _res := _solnproc(_xout); if _xout = "parameters" then return [seq(_pars[_i] = _res[_i], _i = 1 .. nops(_pars))] elif _xout = "initial_and_parameters" then return [seq(_vars[_i+1] = [_res][1][_i+1], _i = 0 .. _n), seq(_pars[_i] = [_res][2][_i], _i = 1 .. nops(_pars))] else return [seq(_vars[_i+1] = _res[_i+1], _i = 0 .. _n)] end if elif type(_xout, `=`) and member(lhs(_xout), ["initial", 'initial', "parameters", 'parameters', "initial_and_parameters", 'initial_and_parameters', NULL]) then _xout := convert(lhs(x_dverk78), 'string') = rhs(x_dverk78); if type(rhs(_xout), 'list') then _res := _solnproc(_xout) else error "initial and/or parameter values must be specified in a list" end if; if lhs(_xout) = "initial" then return [seq(_vars[_i+1] = _res[_i+1], _i = 0 .. _n)] elif lhs(_xout) = "parameters" then return [seq(_pars[_i] = _res[_i], _i = 1 .. nops(_pars))] else return [seq(_vars[_i+1] = [_res][1][_i+1], _i = 0 .. _n), seq(_pars[_i] = [_res][2][_i], _i = 1 .. nops(_pars))] end if elif type(_xout, `=`) and member(lhs(_xout), ["eventdisable", 'eventdisable', "eventenable", 'eventenable', "eventfired", 'eventfired', "direction", 'direction', NULL]) then return _solnproc(convert(lhs(x_dverk78), 'string') = rhs(x_dverk78)) elif _xout = "solnprocedure" then return eval(_solnproc) elif _xout = "sysvars" then return _vars end if; if procname <> unknown then return ('procname')(x_dverk78) else _ndsol; _ndsol := pointto(_dat[2][0]); return ('_ndsol')(x_dverk78) end if end if; try _res := _solnproc(_xout); [seq(_vars[_i+1] = _res[_i+1], _i = 0 .. _n)] catch: error  end try end proc

(6)

dsol3(500)

Warning,  computation interrupted

 

NULL

NULL

``


 

Download dver.mw

 

how to evaulate the value of R(z) for different values of z=0 to 1 with an interval of 0.1 and print ten values  in one column

R(z):= 1-cos^2*(Pi*z);
 

Hi 
I am trying to expand a function f(t) in terms of fractional power series:
for example please see attached file tree.mw
 

``

restart

f := proc (x) options operator, arrow; exp(x) end proc:

alpha := 1/2:

N := 10:

f_approximate := proc (x) options operator, arrow; sum(a[i]*x^`i&alpha;`, i = 0 .. N) end proc


``


 

Download tree.mw

 

thanks 
    

First 438 439 440 441 442 443 444 Last Page 440 of 2133
´╗┐